AFFINE TENSORS IN SHELL THEORY
G´ery De Saxc´e
Laboratoire de M´ecanique de Lille, Universit´e de Lille-I, France e-mail: gery.desaxce@univ-lille1.fr
Claude Vallee
Laboratoire de Mod´elisation M´ecanique et de Math´ematiques Appliqu´ees Universit´e de Poitiers, France
e-mail: vallee@l3ma.univ-poitiers.fr
Resultant force and moment are structured as a single object called the torsor. Excluding all metric notions, we define the torsors as skew-symmetric bilinear mappings operating on the linear space of the affine vector-valued functions. Torsors are a particular family of affine tensors. On this ground, we define an intrinsic differential operator called the af-fine covariant divergence. Next, we claim that the torsor field characteri-zing the behavior of a continuous medium is affine covariant divergence free. Applying this general principle to the dynamics of three-dimensional media, Euler’s equations are recovered. Finally, we investigated more thoroughly the dynamics of shells. Using adapted coordinates, this ge-neral principle provides a consistent way to obtain new equations with non-expected terms involving Coriolis’s effects and the time evolution of the surface.
Key words: tensorial analysis, continuum mechanics, dynamics of shells
1. Introduction
Our starting point is closely related to a new setting developed in mecha-nics by Souriau (1992, 1997a) on the ground of two key ideas: a new definition of torsors and the crucial part played by the affine group of Rn. This group forwards on a manifold an intentionally poor geometrical structure. Indeed, this choice is guided by the fact that it contains both Galileo and Poincar´e
groups (Souriau, 1997b), that allows to involve the classical and relativistic mechanics at one go. This viewpoint implies that we do not use the trick of the Riemannian structure. In particular, the linear tangent space cannot be identified to its dual one and tensorial indices may be neither lowered nor raised.
To each group corresponds a class of tensors. The components of these tensors are transformed according to the action of the considered group. The standard tensors discussed in the literature are those of the linear group of Rn. We will call them linear tensors. A fruitful standpoint consists in considering the class of the affine tensors, corresponding to the affine group. To each group a family of connections allowing one to define covariant derivatives for the cor-responding classes of tensors is associated. The connections of the linear group are known through Christoffel’s coefficients. They represent, as usual, infinite-simal motions of the local basis. From a physical viewpoint, these coefficients are force fields such as gravity or Coriolis’s force. To construct the connection of the affine group, we need Christoffel’s coefficients stemming from the linear group and additional ones describing infinitesimal motions of the origin of the affine space associated with the linear tangent space.
2. Affine tensors
Notations
Let T be a linear space (or vector space) of the dimension n, and (→
eα)
be a basis of T . The associated co-basis (←
eα) is such that ← eα(→ eβ) = δβα. A new basis → eα′ = Pαβ′ →
eβ can be defined through the transformation matrix
P = (Pαβ′). We denote the inverse matrix P−1.
Tensor class
In a previous paper (de Saxc´e, 2002), we proposed a generalization of the usual concept of the tensor, relevant for the mechanics: the tensors are objects the components of which are changed by a given group of transformations (more precisely, they are changed by a linear representation of the considered group). Considering the linear group GL(n), we recover the class of linear tensors. Nevertheless, other choices of transformation groups are possible. In many applications, people customarily handle the orthogonal group O(n), a subgroup of GL(n), that leads to the class of the Euclidean tensors. On
the other hand, considering the affine group A(n), an extension of GL(n) obtained by adding the translations, we define the class of affine tensors.
Affine space
To define an origin Q of the affine space AT associated to T , we can use the column vector V0 collecting the components V0α, in the basis (
→
eα), of the
vector −Q0 joining Q to the zero of T considered as a point of A→ T. By the choice of this affine frame r = (V0, (→eα)), any point V of AT can be identified
to the vector −QV = V−→ α→
eα. Now, let r′ = (V0′, ( →
eα′)) be a new affine frame
of the origin Q′ and basis (→
eα′). Let C′ be the column vector collecting the
components Cα′ of the translation −−→Q′Q in the new basis. The set of all affine
transformations a = (C′, P ) is the affine group A(n). The transformation law
for the affine components Vβ of the point V is
Vα′ = Cα′ + (P−1)αβ′ Vβ (2.1)
Affine functions
Any affine mapping ψ from AT into R is called an affine function of AT. The affine mapping ψ is represented in an affine frame r by
ψ(V ) = χ + ΦαVα
where χ = ψ(Q) and Φα are the components, in the co-basis (←−eα), of the
unique covector ←Φ associated to ψ. We will call (Φ− α, χ) the affine components
of ψ. After a change of the affine frame, they are given by Φα′ = ΦβPαβ′ χ′= χ − ΦβP
β α′Cα
′
as it can be easily verified. The set A∗
T of such functions is a linear space of
the dimension (n + 1).
Vector-valued torsors
Let then R be a linear space of the dimension p ¬ n. We call the torsor any bilinear skew-symmetric mapping (ψ,ψ) 7→b −→µ (ψ,ψ) ∈ R from Ab ∗
T × A∗T
into R. Following Souriau (1992) all the tensorial indices related to R will be located at the left hand of the tensor. With respect to an affine frame r = (V0, (→eα)) of AT and a basis (ρ−→η ) of R, the torsor is represented by
− → V =−→µ (ψ,ψ) =b γµ(ψ,ψ)b γ−→η = γJαβ Φ α Φbβ +γTα(χΦbα−χΦb α) γ−→η
with γJαβ = −γJβα. Let r′ = (V′ 0, (
→
eα′)) be a new frame of AT and γ′−→η = ργ′Qρ−→η be a new basis of R. The corresponding transformation law
is found to be γ′ Tα′ =γρ′(Q−1) (P−1)αµ′ ρTµ (2.2) γ′ Jα′β′ =(P−1)αµ′ (P−1)νβ′ ρJµν+ Cα′ (P−1)βµ′ ρTµ− − (P−1)α′ µ ρTµ Cβ′γρ′(Q−1)
Proper frames and intrinsic torsors
An affine frame will be called a proper frame if the zero vector of T is taken as the origin of the tangent affine space AT : V0 = 0. Any change of
proper frames is a linear transformation (no translation C′ = 0). Restricting
the analysis to linear transformations, we define the class of intrinsic torsors. For any intrinsic torsor −→µ0, transformation law (2.2) degenerates into
γ′
Tα′ =γρ′(Q−1) (P−1)αµ′ ρTµ
γ′
Jα′β′ =γρ′(Q−1) (P−1)αµ′ (P−1)βν′ ρJµν
The components γTα are clearly components of a linear vector-valued tensor
given by a linear mapping from T∗ into R that we call the linear momen-tum. The components γJαβ of the intrinsic torsor can be interpreted as the components of a linear vector-valued tensor given by a bilinear mapping from T∗× T∗ into R that we call the intrinsic angular momentum or spin (Misner
et al., 1973). The intrinsic tensor being given, the affine components of −→µ in any other affine frame r′ = (V′
0, ( →
eα′)) are deduced from general
transforma-tion law (2.2). Thus, γ′Jα′β′ is obtained as the sum of the component of the spin and of the additional term
γ′
JCα′β′ =Cα′ (P−1)βµ′ ρTµ− (P−1)αµ′ ρTµCβ′γ
′
ρ(Q −1)
called the orbital angular momentum (Misner et al., 1973). In conclusion, there is a one-to-one correspondence −→µ0 7→ −→µ between the intrinsic torsors and
3. Affine connection
Affine tangent space
While the difference of the components of two points of an affine space is defined without ambiguousness, the difference of the coordinates of two points in a manifold has no meaning in general. To get round this difficulty, the key idea is to consider that AT is the affine space associated to the tangent linear space TXM, denoted ATXM and called the affine tangent space at X. As
observed by Cartan (1923): ”The affine space at point m could be seen as the manifold itself that would be perceived in an affine manner by an observer located at m”.
Linear connection as sliding
Let X be a point of a manifold M and X′ = X + dX be another point in the vicinity of X. Let us denote the zero vector at X as 0 and at X′
as 0′. For constructing a linear connection, we need to compare the tangent
linear spaces at X and X′ by a suitable identification. A linear space is the corresponding affine space with the zero vector as the particular origin Q. Hence, a linear connection is obtained by a smooth sliding on the manifold of the origin Q = 0 of ATXM onto the origin Q′ = 0′ of ATX′M, as depicted in
Figure 1. Infinitesimal motions of the basis are specified through the connection ∇→
eα= ωαβ →
eβ.
Fig. 1. Linear connecting (sliding)
Affine connection as rolling
On the other hand, let us consider a rolling of the affine tangent space ATXM and let us work on proper frames (Q = 0, Q′ = 0′). The identification
Fig. 2. Affine connection (rolling with initial origin at 0)
Fig. 3. Affine connection (rolling with arbitrary origin)
Fig. 4. Calculation of the affine connection by identifying two neighboring affine tangent spaces
with the neighboring affine tangent space ATX′M shifts the origin Q = 0 onto
a distinct point from Q′ = 0′, as shown in Figure 2. Working more generally in arbitrary affine frames, an affine connection is constructed by rolling of ATXM
with the origin Q shifted onto a distinct point from the origin Q′ of AT X′M,
according to Figure 3. We define the affine connection ωαC as the components of the infinitesimal displacement −dQ =→ −−→QQ′ of the origin when identifying
both neighboring affine tangent spaces (Figure 4). Because of rolling, it holds −→
00′ = dXα→
eα
Hence, the displacement is decomposed as follows ωCα→ eα = −−→ QQ′ =−Q0 +→ −00→′+−−→0′Q′ = (V0α+ dXα) → eα− Vα ′ 0 → eα′ = = dXα→ eα− ∇(V0α → eα) that gives ωαC = dXα− ∇V0α (3.1)
In short, the affine connection provides a smooth variation of the moving affine frames
X 7→ r(X) = V0(X), (→eα(X))
The usual connection matrix ωβαgives a smooth variation of the basis, while the ωα
C specify the motion of the affine space origin. The affine connections
are due to Cartan (1923). In the present section, the key ideas are explained for readers interested in the mechanical science but who are not necessarily aware of advanced concepts of the differential geometry. A presentation using the geometry of principal bundles can be found in (de Saxc´e, 2002) but, in any case, the final result is the same, whether it is obtained by the principal bundle theory or as before.
Affine covariant derivative
On this ground, we are able to calculate the intrinsic covariant derivative of the affine tensors of any type, that we call the affine covariant derivative and denote ∇ (in opposition to the usual covariant derivative ∇ which shoulde be called the linear covariant derivative). First of all, let us consider a field of the tangent vector. As a member of the linear space TXM, it has a linear
covariant derivative, but as a member of the associated affine space ATXM, it
frame r = (V0, (→eα)), any point V of ATXM can be identified to the vector
−−→
QV = Vα→
eα. Thus, its linear covariant derivative is
∇−→V =−−→Q′V′−QV =−−→ −−→Q′Q +−−→QV′−−QV−→
The difference between the two last terms represents the infinitesimal va-riation of the field V as point of the affine space, i.e. its affine covariant derivative. Hence, we obtain the relation
∇−→V = (−ωCα+∇Ve α)→
eα
from which one we deduce
e
∇Vα= ∇Vα+ ωCα (3.2)
Next, we calculate the affine covariant derivative of the affine functions. According to the rule of differentiating a product, one has
e
∇(ψ(V )) =∇(χ + Φe αVα) =∇χ + (e ∇Φe α)Vα+ Φα(∇Ve α)
As the components Φα represent a linear object, the covector
←−
Φ associated to ψ, its affine derivative is just its linear one. Owing to equation (3.2), it holds
e
∇(ψ(V )) =∇χ + (∇Φe α)Vα+ Φα(∇Vα+ ωCα)
Hence, one has
e
∇(ψ(V )) =∇χ + ∇(Φe αVα) + ΦαωCα =∇χ + ∇(ψ(V )) − ∇χ + Φe αωCα
On the other hand, for any field V , we have to satisfy
e
∇(ψ(V )) = d(ψ(V )) = ∇(ψ(V ))
Finally, the affine covariant derivatives of the components of ψ are given by
e
∇Φα = ∇Φα ∇χ = ∇χ − Φe αωαC (3.3)
4. Affine covariant divergence of vector-valued torsors
Thin body
Let M be a manifold of the dimension n representing the physical space in statics (n = 3) and the space-time in dynamics (n = 4). The mapping
N → M : ξ 7→ X = f (ξ) defines a sub-manifold of the dimension p enable one to represent three-dimensional bodies (p = n) or thin ones (p < n). In the sequel, R will be the tangent space TξN at ξ, while AT will be the affine
tangent space ATXM, that is the tangent space TXM at X = f (ξ) endowed
with the structure of the affine space. By a choice of the coordinate systems (Xα) on M and (ξβ) on N , the tangent mapping to f is given by
βUα=
∂Xα
∂ξβ (4.1)
Linear covariant divergence
It is assumed that N is equipped with a symmetric connection
α
βω = αρβγdξρ, where αρβγ = αβργ are Christoffel’s connection coefficients. The
covariant derivative of the tangent vector field −→V =γV γ−→η on N is given by
∇γV = dξβ β∇γV β∇γV =
∂γV
∂ξβ + γ
βργ ρV (4.2)
The manifold M is equipped with a symmetric connection
ωαβ = Γρβα dXβ (4.3)
using Christoffel’s connection coefficients Γα
ρβ = Γβρα. The covariant derivative
of any covector field ←Φ = Φ− α←eα on M is
∇Φα = dXβ ∇βΦα ∇βΦα=
∂Φα
∂Xβ − Γ ρ βαΦρ
Considering the restriction to the sub-manifold N , it holds
γ∇Φα = ∂Xβ ∂ξγ ∇βΦα= ∂Φα ∂ξγ −γU β Γρ βαΦρ (4.4)
Now, we consider a tensor field ξ 7→ T (ξ) of the components γTαas defined before. We hope to calculate its linear covariant derivative, namely γ∇γTα.
According to the rule of differentiating a product, one has for any covector field on M of the components Φα
γ∇(γTαΦα) = (γ∇γTα)Φα+γTα(γ∇Φα)
The left hand member, representing the divergence of a vector field on N , can be developed using (4.2), while, in the right hand member, the last term is transformed owing to (4.4). After simplification, it remains
∂γTα
∂ξγ Φα+ γ
In the last term, replacing α, β, ρ in turn by β, ρ, α, we obtain (γ∇γTα)Φα = ∂γTα ∂ξγ + γ γργ ρTα+γTβ γUρΓρβα Φα
With the covector field Φα being arbitrary, the previous relation is satisfied if
and only if γ∇γTα= ∂γTα ∂ξγ + γ γργ ρTα+γTβ γUρΓρβα (4.5)
This formula allows one to calculate the linear divergence of this class of tensors.
Affine covariant divergence
Now, we are able to calculate the affine covariant derivative of a vector-valued torsor. For any covector field ←F =− γF γ←−η on N , we have
e
∇←F− −→µ (ψ,ψ)b =∇eγF γµ(ψ,ψ)b =∇e γJαβ ΦαΦbβ+γTα(χΦbα−χΦb α)γF
As the affine derivatives of the components γF , Φα,Φbβ and γTα representing
linear objects are equal to their linear derivatives, it holds, according to the rule of differentiating products
e ∇←F− −→µ (ψ,ψ)b =h(∇e γJαβ)ΦαΦbβ+γJαβ∇(ΦαΦbβ) + (∇γTα)(χΦbα−χΦb α)+ +γTα(∇χ)e Φbα− (∇eχ)Φb α +γTαχ(∇Φbα) −χ(∇Φb α) i γF +γµ(ψ,ψ)∇b γF
Taking into account expression (3.3) of the derivative of the affine functions, we obtain after some rearrangements
e
∇←F− −→µ (ψ,ψ)b =h(∇e γJαβ)ΦαΦbβ+γJαβ ∇(ΦαΦbβ) +
+∇γTα(χΦbα−χΦb α)+ (γTα ωβC− ωαC γTβ) i
γF +γµ(ψ,ψ)∇b γF
which can be simplified as follows
e ∇←F− −→µ (ψ,ψ)b =h(∇e γJαβ)ΦαΦbβ+ ∇ γµ(ψ,ψ)b − (∇γJαβ)ΦαΦbβ+ +(γTα ωβC− ωα C γTβ)ΦαΦbβ i γF +γµ(ψ,ψ)∇b γF
On the other hand, because the value of ←F for− −→µ is a scalar field, we have
e
∇←F− −→µ (ψ,ψ)b = d←F− →−µ (ψ,ψ)b = ∇←F− −→µ (ψ,ψ)b = = ∇ γµ(ψ,ψ)b γF +γµ(ψ,ψ)∇b γF
Hence, for any ψ, ψ andb ←F , it holds−
e ∇γJαβ− ∇γJαβ − ωCα γTβ +γTαωβ C γF Φα Φbβ = 0
With the affine functions and covectors being arbitrary, we obtain an expres-sion of the affine covariant derivative of the vector-valued torsors
e
∇γTα= ∇γTα ∇e γJαβ = ∇γJαβ+ ωαC γTβ−γTα ωβC (4.6) By analogy with (4.3), we introduce the affine connection coefficients Γα
ρC
such that
ωCα = ΓρCα dXρ= ΓρC γα Uρdξγ Owing to (3.1), one has
ΓρCα = δρα− ∂V α 0 ∂Xρ − Γ α ρβV0β (4.7)
As a particular case of (4.6), we obtain the affine covariant divergence of a vector-valued torsor
γ∇e γTα =γ∇γTα
(4.8)
γ∇e γJαβ =γ∇γJαβ+γUρΓρCα γTβ−γTαγUρΓρCβ
where the divergence of γTα is given by (4.5) and γ∇γJαβ =
∂γJαβ
∂ξγ + γJρβ
γUµΓµρα +γJαργUµΓµρβ +γγργ ρJαβ (4.9)
which can be easily obtained by reasoning as for the proof of (4.5).
5. Dynamics of three-dimensional bodies
Three-dimensional bodies
Let a continuous medium (a solid or a fluid) occupying an open domain Ω ⊂ R3 that we call also a body. In order to model its evolution
f : ]t0, t1[×Ω → M. For convenience, we choose the same coordinate
sys-tem on N and M. Hence, the local expression of f is the identity mapping Xα = ξα. The distinction between the left and right hand indices becomes ir-relevant and, in the present section, we put all the indices at the right hand as usual. Thus, we have βUα= δαβ and γ∇ =e ∇eγ. Moreover, we write γTα= Tαγ
and γJαβ = Jαβγ. The behavior of the continuous medium is described by a
torsor field X 7→ −→µ (X). We claim that the balance (or conservation law) of momentum of the continuous medium says that the torsor field is affine covariant divergence free
e
∇γTαγ= 0 ∇eγJαβγ = 0 (5.1)
Following Souriau (1992, 1997a), the first equation traduces the balance of linear momentum. The second one is a full covariant version of the balance of angular momentum as presented by Misner et al. (1973, p. 156).
Balance of the angular momentum
Three-dimensional continuous media are mainly considered in the litera-ture as non polarized media, according to Cauchy’s famous theory.
Let (Tαγ, Jαβγ) be the affine components of the unique intrinsic torsor
field X 7→ −→µ0(X), associated to X 7→ −→µ (X), in a moving proper frame
X 7→ r(X) = (0, (→
eα(X)). We claim that the intrinsic torsor is spin free
Jαβγ = 0. Thus, accounting for (4.7), (4.8)
2, the balance of angular momentum
(5.1)2 leads to
e
∇γJαβγ = Tβα− Tαβ = 0
In this symmetry condition of the linear momentum, the reader can clearly recognize the classical hypothesis of Cauchy’s media.
Galilean tensors
All what has been said so far may be applied as much for the general relativity theory as for the classical mechanics. Henceforth, we shall restrict the analysis to the latter theory. In the sequel, Greek indices are 0 to 3 while Latin ones run from 1 to 3 (associated to the space coordinates only). Any point X of the space-time M represents an event occurring at position r and time t. With an appropriate coordinate system, it is represented by Xi = ri, X0 = t.
the Galilean transformations, that is the affine transformations a = (C′, P )
such that (Souriau, 1997b)
C′= " τ k # P = " 1 0 u R #
where u ∈ R3is a Galilean boost, R ∈ SO(3) is a rotation, k ∈ R3is a spatial translation and τ ∈ R is a clock change. Any coordinate change representing a rigid body motion and a clock change
r′= (R(t))⊤ r − r0(t)
t′ = t + τ0
where t 7→ R(t) ∈ SO(3) and t 7→ r0(t) ∈ R3 are smooth mappings and
τ0 ∈ R is a constant, is called a Galilean coordinate change. Indeed, the
cor-responding Jacobean matrix is a linear Galilean transformation
P = ∂X ′ ∂X = " 1 0 u R #
where u = ̟(t) × (r − r0(t)) + ˙r0(t), is the well-known velocity of transport.
It involves Poisson’s vector ̟ such that ˙R = j(̟)R, where j(̟) (sometimes also denoted ad(̟)) is the skew-symmetric matrix representing the cross-product by ̟ : ∀v ∈ R3, j(̟)v = ̟ × v.
Galilean connections
At each group of the transformation G a family of connections and the corresponding geometry (called the G-structure by Dieudonn´e (1971)) is as-sociated. We call Galilean connections the symmetric connections associated to Galileo’s group. In a Galilean coordinate system, they are given by
ω = " 0 0 j(Ω)dr − gdt j(Ω)dt # (5.2)
where g is a column-vector collecting the gj = −Γ00j and identified to the
gravity (Cartan, 1923), while Ω is a column-vector associated by the mapping j−1 to the skew-symmetric matrix the elements of which are Ωi
j = Γj0i and
Balance of the linear momentum
In the present sub-section, we shall follow the reasoning proposed by Souriau (1992, 1997a). Let be an Eulerian representation of the continuous medium in which any event is represented in a Galilean coordinate system by Euler’s coordinates Xi = ri and X0 = t. On the other hand, let be a
La-grangean representation in which the same event is represented by Lagrange’s coordinates of the material particle Xi′ = si′ and X′0= t′, which are not in
general Galilean coordinates. They are related to the previous representation by a smooth coordinate change like
ri= ϕi(sj′, t′) t = t′ We introduce the deformation gradient Fi
j′ = ∂ri/∂sj ′
, and the velocity ui= ∂ri/∂t.
The particle of the coordinates (sj′
) being at rest in the Lagrangean re-presentation has the trajectory such that dsj′ = 0. Differentiating, we obtain
dX = " dt dr # = " 1 0 u F # " dt′ ds # = P dX′ (5.3)
To adjust to usual convention in the continuum mechanics (compressive stres-ses are negative), we put Si′j′ = −Ti′j′. The components Si′j′ are generally recognized as representing the internal forces or stresses in the Lagrangean representation, and are customarily called symmetric Piola-Kirchhoff stresses. The component ρ = T′00can be interpreted as the mass density. The particles being at rest in this particular representation, the components T0i′ = Ti′0,
interpreted as the linear momentum, are supposed to vanish. In Euler’s coor-dinates, it holds, owing to (2.2)1
T = P T′P⊤= " 1 0 u F # " ρ 0 0 −S # " 1 u⊤ 0 F⊤ # = " ρ ρu⊤ ρu −σ + ρuu⊤ # (5.4) where σk = ρuu⊤ collects kinetic stresses and, according to Simo (1988)
σ = F SF⊤ collects Cauchy’s stresses. In Euler’s representation, the events
being given by a Galilean coordinate system, the connection matrix is given by (5.2). Thus, accounting for (4.8)1, equation (5.1)1 can be interpreted as
Euler’s equations of the continuous medium ∂ ∂rj(ρu j) + ∂ρ ∂t = 0 ρ∂u i ∂t + u j∂ui ∂rj = ∂σ ij ∂rj + ρ(g i− 2Ωi juj)
6. Adapted coordinates
Moving surface
We want to model problems of the dynamics of shells and plates within the frame of the classical mechanics. We have to represent the time evolution of a smooth material surface.
Hence, we are in the case n = 4 and p = 3 < n. In the sequel, the indices a, b, c, e associated to the surface parameters takes only values 1 or 2, while the other Latin indices such that i, j, k are running from 1 to 3 as before. In the Eulerian representation, let us consider a Galilean coordinate system (Xα) on
the space-time M. Interpreting the last coordinate ξ0 on the submanifold N
as the time, let us suppose that the mapping N → M : ξ 7→ X = f (ξ) is represented in local coordinates by given equations
Xi= ri= pi(ξγ) = pi(ξa, ξ0) = pi(ξa, t) X0 = t = ξ0
Adapted coordinates
A classical tool of the theory of surfaces is the tangent plane to the current point. In order to separate in-plane and off-plane components of the torsor and the balance of momentum, we introduce another coordinates (Xβ′
) of the space-time. The new spatial coordinates are X′a, denoted θa, and X′3, denoted θ3. The time coordinate X′0, denoted t′, is unchanged. The
key-idea is to choose the new coordinates in such a way that the equation of the material surface at a given time is merely
θ3= 0
In these adapted coordinates, the local representation ξ 7→ X′ of the
map-ping f defining the sub-manifold N is
θa= ξa θ3= 0 t′ = ξ0 (6.1)
The adapted coordinates are related to the previous ones through equations like
ri= pi(θa, t) + θ3ni(θa, t) t = t′ (6.2) For convenience, following for instance (Naghdi, 1972), we choice ni such that
3 X i=1 πaini= 0 3 X i=1 nini = 1 (6.3)
with πia= ∂p i ∂θa v i = ∂pi ∂t β i a= ∂ni ∂θa w i = ∂ni ∂t
Let n be a column vector of the components ni and π (resp. β) be a matrix the element of which at the ath row and ith column is πai (resp. βai). As usual, n is interpreted as the unit vector normal to the material surface and π as the projector onto the tangent plane (to the material surface at the current time) (Valid, 1995). The uniqueness of n is ensured by conditions (6.3). The column vector v, collecting the components vi, represents the velocity and the column vector w, collecting the components wi, represents the time rate
of the unit normal vector. Hence equations (6.3) read
πn = 0 n⊤n = 1 (6.4)
Differentiating (6.4)2 leads to n⊤dn = 0. Thus
βn = 0 n⊤w = 0 (6.5)
Calculation of the connection matrix
Notice that the new coordinates (Xβ′) are not generally Galilean. Hence the connection matrix ω′in this coordinate system does not have the standard
form of (5.2). Differentiating (6.2) gives successively P = " 1 0 0 v + θ3w π⊤+ θ3β⊤ n # (6.6) dP = " 1 0 0 dv + wdθ3+ θ3dw π⊤+ β⊤dθ3+ θ3dβ⊤ dn #
Putting θ3= 0 on the material surface, leads to
P = " 1 0 0 v π⊤ n # dP = " 1 0 0 dv + wdθ3 π⊤+ β⊤dθ3 dn # (6.7) Owing to (6.4), the inverse transformation matrix is
P−1 = 1 0 −vt a−1π −v3 n⊤ (6.8)
where, the symmetric matrix a = ππ⊤, represents the first fundamental form
of the material surface, vt = a−1πv is the in-plane velocity and v3 = n⊤v is
the off-plane component of the velocity. Because of the classical transformation law
ω′ = P−1ωP + P−1dP
and taking into account (5.2) and (6.7-8), the connection matrix in the adapted coordinate system is ω′ = 0 0 0 a−1πA1 a−1π(A2+ β⊤dθ3) a−1π(dn + j(Ω)ndt) n⊤A1 n⊤A2 0 (6.9) where A1= j(Ω)(dr + vdt) − gdt + dv + wdθ3 A2= dπ⊤+ j(Ω)π⊤dt
Its elements have to be expressed with respect to the differential of the adapted coordinates dθi and dt′. For convenience, we introduce the column vector dθt collecting dθa, and we put dt′ = dt. By introducing the linear
operators dθt = dθ a ∂ ∂θa d dt = I3 ∂ ∂t + j(Ω) where I3 is the 3 × 3 identity matrix, we have
dπ⊤+ j(Ω)π⊤dt = dθtπ ⊤+dπ⊤ dt dt dn + j(Ω)ndt = dθtn + dn dtdt
Let gp be a column vector representing the acceleration of transport
gp= ∂v ∂t = ∂2p ∂t2 and ∂v ∂θt = ∂ 2p ∂θt∂t = ∂π ⊤ ∂t Accounting for (6.5), it holds
j(Ω)(dr + vdt) − gdt + dv + wdθ3= j(Ω)(π⊤dθ t+ ndθ3+ 2vdt) − gdt + +gpdt + ∂π⊤ ∂t dθt+ ∂n ∂tdθ 3 = dπ ⊤ dt dθt+ dn dtdθ 3− g∗ dt
where g∗ = g − 2Ω × v − g
p, is the gravity in the adapted coordinate system,
obtained by subtracting Coriolis’s acceleration gc = 2Ω × v and the
acce-leration of transport gp from the gravity in the Galilean coordinate system.
Because of (6.5)2, one has
n⊤dn dt = n
⊤w + n⊤j(Ω)n = 0
The connection matrix (6.9) becomes
ω′= 0 0 0 a−1πA3+ dn dtdθ 3 a−1πA 4+ β⊤dθ3 a−1πdθtn + dn dtdt n⊤A 3 n⊤A4 0 (6.10) where A3= dπ⊤ dt dθt− g ∗dt A 4 = dθtπ ⊤+dπ⊤ dt dt
As we shall be working latter on in the adapted coordinate system, for sake of easiness, we cancel the prime symbol. It clearly results from (6.1) that
aUb = δba aU3= 0 0Ui= 0 aU0 = 0 0U0 = 1
(6.11)
7. Shell variables
Balance of momentum
Let a continuous medium of an arbitrary dimension p ¬ n, the behavior of which is described by a torsor field ξ 7→ −→µ (ξ) on N . Generalizing the approach of Section 5, we claim that the balance of momentum says that this torsor field is affine covariant divergence free
γ∇e γTα = 0 γ∇e γJαβ = 0 (7.1)
Discussion
Any torsor has pn components γTαand, owing to the skew-symmetry with respect to the right hand side indices, pn(n − 1)/2 independent components
γJαβ. Then, it has pn(n + 1)/2 affine components. On the other hand, a torsor
field is subjected to n(n + 1)/2 independent scalar equations (7.1). In the statics of shells (p = 2, n = 3), only 6 equations are available to determine 12 variables. In the dynamics (p = 3, n = 4), the difficulty is higher with 10 equations for 30 variables.
Fortunately, it is possible in the Galilean setting to reduce the redundancy of the shell by introducing additional hypothesis related to the modeling. A resisting material surface can be seen as an approximation of a three dimensio-nal medium, as a consequence of the fact that it is thin in the normal direction to the middle surface. There is a broad variety of situations such as a smooth curved sheet or a composite laminate but also a smooth surface approximating a corrugated sheet, a lattice or a fluid moving between two close sheets and so on. Although the general modeling proposed as before is relevant to represent this wide range of situations, it would be a heavy task to examine every one of them. Hence, we only wish to illustrate our method by focussing the attention on the most simple case of an homogeneous curved thin sheet of the current thickness h.
The behavior of a three dimensional body is characterized by a spin free torsor field with components Tαγ, as discussed in Section 5. We hope to build
a shell torsor field with the affine components γTα and γJαβ by a suitable
integration over the thickness. The shell variables γTα, γJαβ are related to an infinitesimal surface element modeling a piece of the three dimensional body occupying the volume over and above the surface element in the thickness direction, and called the shell element.
Three dimensional continuum torsor
In a first draft, we adopt two usual hypotheses. On the shell element scale, the surface curvature is neglected and the strain is small (but not necessarily the displacements and rotations). A more sophisticated method using the con-cept of the affine transport is proposed in (de Saxc´e, 2002a), but we shall not follow this way in the present work. According to the approach of Section 5, let Xi′ = si′ be Lagrange’s coordinates of the material particles of the three dimensional continuum and X′0 = t′. Neglecting the strains, the motion of
the shell element can be locally approximated by a rigid motion r = ϕ(s, t) = p(t) + R(t)s t = t′
where p(t) ∈ R3 represents the position of the point on the middle surface (in
short, the middle point) and the time dependent rotation matrix R(t) ∈ SO(3) describes the rigid motion of the shell element around this point. Relation (5.3) degenerates into dX = " dt dr # = " 1 0 u R # " dt′ ds # = P dX′
involving the velocity of transport of the middle point v = ˙p and of any point of the shell element
u = v + ̟ × (r − p)
where ̟ is Poisson’s vector ˙R = j(̟)R. Taking into account (6.2), it holds
u = v + θ3̟ × n (7.2)
On the other hand, (6.6) says that
u = v + θ3w (7.3)
Identifying (7.2) to (7.3), leads to w = ∂n
∂t = ̟ × n (7.4)
In the Eulerian representation, the torsor components Tαγ are given by
(5.4). On the considered scale, the surface curvature effects can be neglected. The inverse transformation matrix is approximated by its value (6.8) on the middle surface. Thus, in the adapted coordinates, the new components Tα′γ′
are given by T′ = P−1T P−⊤= 1 0 −vt a−1π −v3 n⊤ " ρ ρu⊤ ρu −σ + ρuu⊤ # " 1 v⊤t −v3 0 (a−1π)⊤ n #
Accounting for (6.5) and (6.2)-(6.3), it holds
T′= ρ ρθ3(a−1πw)⊤ 0 ρθ3a−1πw a−1π − σ + ρ(θ3)2ww⊤(a−1π)⊤ −a−1πσn 0 −n⊤σ(a−1π)⊤ −n⊤σn
Therefore, in the adapted coordinates, the stress components are σ′ab= 3 X i,j=1 aaeπei σij πcj acb σ′33= 3 X i,j=1 niσij nj σ′b3= σ′3b= 3 X i,j=1 aaeπei σij nj while the new velocity components are
w′a=
3 X i=1
aaeπiewi
Canceling the prime symbol for sake of easiness, we obtain the torsor compo-nents of the three dimensional medium in the adapted coordinates
Tab= −σab+ ρ(θ3)2wawb Ta0 = T0a= ρθ3wa
Ta3= T3a= −σa3 T30= T03= 0 T00= ρ (7.5)
This result has been obtained according to the two classical previous hypothe-ses. They could be eliminated by a more pervasive analysis using for instance tools developed in (Hamdouni et al., 1999) by considering a shell as a stacking-up of curve sheets.
Integration over the thickness
If the torsor components are calculated with respect to the current point r of the three dimensional medium in a proper frame, the spin components vanish.
It is recalled that every point X of the manifold M corresponds to an event occurring at a given position and time. If we neglect once again the curvature effects on the shell element scale, the manifold M can be approximated by the affine tangent space ATXM at the current point X. Naturally, we are
working with the basis (→
eα) associated to the considered adapted coordinate
system. The current point X of coordinates (X0 = t, Xa = θa, X3 = θ3) is
identified to the origin of the proper frame, that is zero of the tangent linear space TXM. Concerning the position of the middle point at the same time,
the point X′ of the coordinates (X0 = t, Xa= θa, X3 = 0) is represented by
the point of the affine tangent space ATXM with the affine coordinates
V = t θ1 θ2 0 − t θ1 θ2 θ3 = 0 0 0 −θ3
Let us take this point as a new origin. In the non-proper frame r′ = (V, (→
eα)), the considered point has vanishing coordinates V′ = 0.
Ac-cording to transformation law (2.1), we consider the affine transformation a = (C′, P ) with P being the identity of R4, and
C′ = V′− V = 0 0 0 θ3 (7.6)
Owing to transformation law (2.2), the components of the linear momentum are unchanged while those of the angular momentum do not vanish due to the existence of the orbital angular momentum any longer
Jα′β′γ′ = Cα′Tβ′γ′ − Tα′γ′Cβ′
For easy notations, we cancel the prime symbol. Owing to (7.6), the non-vanishing components of the angular momentum in the new frame are
J3bγ = −Jb3γ = θ3Tbγ J30γ= −J03γ= θ3T0γ (7.7) Under the previous approximations, the torsor at every point of the shell and at the considered time is given by its components with respect to the affine frame associated to the middle point. By integrating them over the thickness, we obtain the shell variables
γTα= h/2 Z −h/2 Tαγ dθ3 γJαβ = h/2 Z −h/2 Jαβγ dθ3 (7.8)
Combining (7.5), (7.7) and (7.8) leads to
aTb = − h/2 Z −h/2 σab dθ3+ρh 3 12 w awb aT3 = − h/2 Z −h/2 σa3 dθ3 0T0 = ρh (7.9) aJb3= −aJ3b = h/2 Z −h/2 θ3σabdθ3 0J3b = −0Jb3= ρh3 12 w b
the other variables being zero. The previous result is neither general nor exact but it illustrates a method to construct the shell variables and provides their physical interpretation. Accounting for the symmetry of Cauchy’s stress tensor, it remains 11 independent non-zero shell variables in the Galilean setting, instead of 30 in the general affine geometry.
8. Usual theory of plates and shells in static equilibrium
Shell variables
Let us assume that all the points of the three dimensional body are at rest in the Galilean coordinate system (Xα) at any time. The function ni
does not explicitly depend on t = ξ0 and, consequently, the velocity w
va-nishes. The components Nab = −aTb are line densities of membrane forces, and the components Qa = −aT3 are line densities of transverse shear forces.
The component ρs = 0T0 is interpreted as a surface density of mass. The
components Mab = aJb3 can be interpreted as line densities of bending and twisting couples. Shell variables (7.9) are reduced to
Nab = −aTb = h/2 Z −h/2 σab dθ3 Qa= −aT3 = h/2 Z −h/2 σa3 dθ3 (8.1) ρs=0T0 = ρh Mab=aJb3= −aJ3b = h/2 Z −h/2 θ3σab dθ3
the other variables are zero. Taking into account the symmetry of the stress tensor, it remains 9 independent variables in statics, instead of 11 in dynamics.
Static equilibrium
Let us assume that all the points of the material surface are at rest in the Galilean coordinate system (Xα) at any time. The functions ridoes not expli-citly depend on t = ξ0 and, consequently, the velocity v and the acceleration
of transport gp vanish. Besides, we suppose that Coriolis’s effects are absent
Ω = 0. Therefore, it holds dπ⊤ dt = 0 dn dt = 0 g ∗ = g
Connection matrix (6.10) becomes
ω′= 0 0 0 −gtdt a−1πdθtπ⊤+ β⊤dθ 3 a−1πd θtn −gndt n⊤d θtπ⊤ 0
with: gt = a−1πg, g3 = n⊤g. Moreover, we assume that the connection on
of the matrix a−1 at the bth row and eth column will be denoted abe. For
convenience, we put
cai = aaeπei
The following Christoffel’s coefficients are generated Γbca =abcγ = cai∂π i b ∂θc Γ a 3b = cai ∂ni ∂θb = −b a b Γ00a = −caigi = −gta (8.2) Γ3 ab= 3 X i=1 ni∂π i b ∂θa = bab Γ 3 00= −g3
with the other ones being zero. As usual, the coefficients babdefine the 2nd
fun-damental form of the surface.
Let us examine the particular form of the balance of linear momentum (7.1)1 in the adapted coordinates. Accounting for (6.11), (8.1) and (8.2), many
terms disappear. For the in-plane translation equilibrium, it remains −γ∇γTa= Nba b− b a bQb+ ρsgta= 0 (8.3) where Nba b = ∂Nba ∂θb + Γ a bcNbc+ ΓcbcNba
is the linear covariant divergence with respect the connection on the material surface. Similarly, for the off-plane translation equilibrium equation, it holds
−γ∇γT3 = babNab+ Qb b+ ρsg 3 = 0 (8.4) where Qb b = ∂Qb ∂θb + Γ c bcQb
The last equation provides the balance of mass in the Lagrangean repre-sentation
γ∇γT0 = ∂ρs
∂t = 0 (8.5)
For the balance of angular momentum (7.1)2, we need to evaluate first the
linear covariant divergence by (4.9), next the affine one by (4.8). For the in-plane rotation equilibrium, we have
γ∇γJb3= ∂Mab ∂θa + Γ b acMac+ ΓaccMab= Mab a
Thus, it holds γ∇e γJb3= Mab a− Q b= 0 (8.6)
For the off-plane rotation equilibrium, one has successively
γ∇γJ21= −b1aMa2+ b2aMa1
(8.7)
γ∇e γJ21=γ∇γJ21− N21+ N12= (N12− b1aMa2) − (N21− b2aMa1) = 0
Let us define the symbol εcb such that
ε12= −ε21= 1 ε11= ε22= 0
Equation (8.7) reads
γ∇e γJ21= εcb(Ncb− bcaMab) = 0 (8.8)
which can be recognized as the classical symmetry relation of the usual shell theory (Valid, 1995). We recover the standard system of equilibrium equations of shells (8.3)-(8.6), (8.8) (Green and Zerna, 1968) but as the expression in an adapted coordinate system of the free affine covariante torsor principle (7.1). Nevertheless, these 7 scalar equations are not sufficient to determine the 9 independent shell variables (8.1), and we need additional conditions, the constitutive laws, but this topic will not be treated here.
9. Consistent formulation of the dynamics of plates and shells
Dynamic equilibrium
We start again with general expression (6.10) of the connection matrix in the adapted coordinates. In addition to (8.2), new Christoffel’s coefficients arise, due to both Coriolis’s effects and the time evolution of the surface
Γb0a = cai∂π i b ∂t + Ω i jπjb = Φab Γ30a = cai∂n i ∂t + Ω i jnj = Φa (9.1) Γ3 b0 = 3 X i=1 ni∂π i b ∂t + Ω i jπbj = Φb
According to definitions (8.1), non-zero shell variables (7.9) are aTb = −Nab+ρh3 12 w awb aT3 = −Qa 0T0 = ρh aJb3= −aJ3b = Mab 0 J3b = −0Jb3= ρh 3 12 w b (9.2)
In the adapted coordinates, the balance of linear momentum (7.1)1 gives the
following equations. For the in-plane translation equation, it holds −γ∇γTa=Nba−ρh 3 12 w awb b− b a bQb+ ρscai gi− 2Ωjivj − ρs ∂vi ∂t = 0 (9.3) The off-plane translation equation is
−γ∇γT3 = bab Nab−ρh 3 12 w awb+ Qb b+ 3 X i=1 niρs(gi− 2Ωjivj) − ρs ∂vi ∂t = 0 (9.4) By comparison with corresponding static equilibrium equations (8.3) and (8.4), we obtain additional terms reflecting expected Coriolis’s and inertia effects. Less classical are kinetic terms similar to those arising in the equations of three-dimensional continuous medium given in Section 5. In mechanics of solids, they are generally neglected, but we have to notice that their magnitude could become significant at the high velocity, for instance in the case of impact. Finally, the last equation provides the balance of mass
γ∇γT0 =
∂ρs
∂t + Φ
a
aρs= 0 (9.5)
Next, we calculate the linear covariant divergence of the angular momentum by (4.9). The non-zero components are
γ∇γJ21= −b1aMa2+ b2aMa1+ Φ10J23− Φ20J13 (9.6) γ∇γJb3= ∂γJb3 ∂ξγ + Γ b accJa3+ ΓρccρJb3+ Φba0Ja3
The affine covariant divergence of the angular momentum is given by (4.8). Let us determine the particular form of the balance of angular momentum (7.1)2 in the adapted coordinates. The off-plane rotation equation is
γ∇e γJb3= Mab a− Q b−ρh3 12 ∂wb ∂t + Φ b awa+ Φccwb = 0 (9.7)
The first two terms are the same as in static equilibrium equation (8.6). The third one represents inertia effects and is expected. On the other hand, the last two terms are non-standard in the literature as subtly resulting from the time evolution of the surface geometry through new Christoffel’s coefficients (9.1). Besides, owing to (9.6), the in-plane rotation equation
γ∇e γJ21= −εcb(cTb+ bcaaJb3− Φc0Jb3) = 0
generalizes the symmetry relation of the usual shell theory to the dynamics. Owing to (9.2), one has equivalently
γ∇e γJ21= εcb Ncb− bcaMab−ρh 3 12 (Φ c+ wc)wb= 0 (9.8)
Once again, new terms appear, taking into account kinetic terms and the time evolution of the surface geometry. Finally, it can be seen that the remai-ning equations are automatically satisfied
γ∇e γJ03= 0 γ∇e γJb0= 0
In short, the behavior of the torsor field is governed by a system of 7 scalar equations, namely (9.3)-(9-5) and (9.7)-(9.8). On the other hand, we have 11 independent unknowns, Qb, Nab, Mab, h, wb linked to shell variables (9.2) and 3 additional variables vi. Initially equal to 30 − 10 = 20 in the most
general case, the redundancy degree is now reduced to 14 − 7 = 7. To get rid of this indeterminacy, we must introduce additional assumptions concerning the constitutive laws.
What about the 30 − 11 = 19 other shell variables? Under various assump-tions introduced in Secassump-tions 6 to 9, they are in a way asleep. Nevertheless, their existence is predicted by the general theory within the frame of the affine group geometry. They could be waken up by considering other idealization than the one of a smooth curved thin sheet undergoing small deformation, and above all in a relativistic context.
10. Conclusions
Firstly, although the affine geometry could appear as a poverty-stricken mathematical frame, we think it is sufficient to describe the fundamental tools of the continuum mechanics. It leads to a definition of the torsors which is
completely relieved of all metric features. Next, we developed the correspon-ding Affine Tensor Analysis enabling one to propose a general principle of an affine divergence free torsor. We showed that this principle allows one to recover the balance equations of the statics of three dimensional bodies. For the dynamics of shells, we revealed the existence of new terms, depending on velocities and involving time variation of the surface geometry. Of course, so-me open problems deserve more pervasive investigations. We have to develop at first applications of the new shell theory in order to assess the magnitu-de of the predicted terms. Other subjects of interest would be the dynamics of beams. Finally, we would like to point out related topics. In the previous work, the first author proved that for the dynamics of particles and rigid bo-dies, the well-known theorem of angular momentum is also a consequence of our principle of balance (de Saxc´e, 2002). Besides, there is a subtle link with the symplectic mechanics which leads to a nice extension of Kirillov-Kostant-Souriau theorem.
References
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Tensory afiniczne w teorii powłok
Streszczenie
Strukturę formalna wypadkowej siły i momentu można ująć w postaci pojedyn-czego obiektu zwanego torsorem. Wyłączając wszystkie pojęcia metryczne, torsory definiujemy jako skośno-symetryczne dwuliniowe odwzorowania w przestrzeni linio-wej w dziedzinie funkcji wektorowych. Torsory stanowią szczególną rodzinę afinicz-nych tensorów. Na tej podstawie zdefiniowano wewnętrzny operator różniczkowania zwany afiniczną kowariantną dywergencją. Następnie wysunięto postulat, że zachowa-nie się ośrodka ciągłego opisane polem torsorowym posiada zerową taką dywergencję. Zastosowawszy tę ogólną zasadę, użyto równań Eulera w opisie dynamiki ciał trójwy-miarowych. W dalszej części pracy skoncentrowano się na dynamice powłok. Poprzez użycie odpowiednio zaadaptowanych współrzędnych wykazano, że zastosowanie tej ogólnej zasady stanowi spójną metodę otrzymywania równań z nieoczekiwanie po-jawiającymi się członami odpowiedzialnymi za efekty przyspieszenia Coriolisa oraz zmian powierzchni powłoki w czasie.
